Exploring the landscape of heterotic strings on Td

Abstract

Compactifications of the heterotic string on Td are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in IId,d+16. In agreement with the duality between heterotic on T2 and F-theory on K3, all possible gauge groups on T2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

Cobordism conjecture, anomalies, and the String Lamppost Principle

“…Our constraints are consistent with everything we know in string compactifications. 26 In particular, they are consistent with the results of a systematic exploration of all the possible maximal symmetry enhancements for heterotic on T 2 in [64]. In the eight-dimensional case, there could be a top.…”

“…Thus, we get constraints on the allowed global forms of the gauge group. These are all satisfied in known heterotic examples; see [64] for an exhaustive list, including partial information on the global structure of the gauge group. For instance, there is a point on the 9d moduli space where the Lie algebra is so (34).…”

“…only has real representations, since the fundamentals of both SU(2) and E 7 are pseudorreal, and the quotient identifies them to produce a real representation. This group actually arises in heterotic moduli space by adjoint breaking [64,65], so it better have dim(G) + rank(G) equal to a multiple of eight. And indeed, rank is 8 and dimension is 3+133=136, for a total of 144 which is a multiple of 8.…”

“…This point corresponds to class 297 of[64] and it is just heterotic on T 2 with complex structure τ = e πi/3 and a torus at a self-dual radius. The Z3 corresponds to rotations by 120 • of the T 2 .…”

Four-dimensional N = 1 theories, S-fold constraints on T-branes, and behaviors in IR and UV

“…Discussions of extremal elliptic K3 surfaces in the context of string theory can be found, e.g., in[63][64][65][66][67]120] 20. We follow the convention of the term used in[121] 21.…”